Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #362481 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2293 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13918 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15,19,32)(4,10,24,37)(5,25,27,38)(6,12)(7,14)(8,13)(9,18,33,20)(11,26,39,28)(16,34)(17,44)(21,49)(22,51)(23,36,29,40)(30,48)(31,46)(35,54)(41,47)(42,61,62,55)(43,57)(45,56)(50,64,63,58)(52,59)(53,60);(1,3,6,18,7,19,8,20)(2,9,13,32,14,33,12,15)(4,21,29,16,24,48,23,43)(5,17,26,45,27,46,28,47)(10,34,36,49,37,57,40,30)(11,31,38,56,39,44,25,41)(22,50,54,42,53,63,52,62)(35,58,51,61,59,64,60,55);(1,4,6,23,7,24,8,29)(2,10,12,36,14,37,13,40)(3,16,18,21,19,43,20,48)(5,22,26,52,27,53,28,54)(9,30,15,34,33,49,32,57)(11,35,25,51,39,59,38,60)(17,42,45,50,46,62,47,63)(31,55,41,58,44,61,56,64);(1,5,7,27)(2,11)(3,17,19,46)(4,22,24,53)(6,26,8,28)(9,31)(10,35)(12,25)(13,38)(14,39)(15,41)(16,42,43,62)(18,45,20,47)(21,50,48,63)(23,52,29,54)(30,55)(32,56)(33,44)(34,58)(36,51)(37,59)(40,60)(49,61)(57,64)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269761);